6.5 THREEFIRM CASE: ADDING A RISKFREE ASSET
Suppose you are considering a portfolio of three securities, and you are permitted to borrow and lend at 12%. The portfolio in Table 6.1 provides an expected return of 14%, as derived in Chapter 4, in the topic titled ThreeFirm Case: Application of Mean Variance Theory:
Table 6.1: MinimumVariance Portfolio Weights for an Expected Return = 14%. 
Portfolio 
Weights 
Firm 1 
0.41 
Firm 2 
0.18 
Firm 3 
0.41 
Call this Portfolio I. Now consider the set of portfolios that can be attained by combining Portfolio I with the riskfree asset. Returns from such portfolios are described by the straight line arf + (1 a)E(r) where proportion a of the portfolio is invested in the riskfree security and (1 a) is invested in Portfolio I. The proportion of total wealth that is allocated to risky assets in Portfolio I equals 41% in Firm 1, 18% in Firm 2, and 41% in Firm 3. Suppose your wealth is $150,000, and you allocate $100,000 to the three risky assets and $50,000 to the riskfree asset. When this is the case a is 1/3 and (1a) is 2/3. As a result, $41,000 is invested in Firm 1, $18,000 is invested in Firm 2, and $41,000 is invested in Firm 3. This example provides an expected return equal to (1/3)12%+(2/3)14% = 13 1/3%.
Suppose your target return is 13%. This target can be attained in two ways. First, there exists a point on the minimumvariance (bulletshaped) frontier that provides 13%. In CAPM Tutor it is easy to identify this portfolio by clicking on the appropriate part of the minimumvariance frontier. You may locate the appropriate portfolio this way (by trial and error) or by working through the analytics provided in Chapter 4, topic 4.9, titled ThreeFirm Case: Application of Mean Variance Theory:
Firm 1 = 6.288E  0.460
Firm 2 = 1.153E + 0.338
Firm 3 = 5.075E + 1.122
By letting E be 0.13, you can compute the resulting portfolio weights in Table 6.2.
Table 6.2: MinimumVariance Portfolio Weights 
for an Expected Return = 13%. 
Portfolio 
Weights 
Firm 1 
0.349 
Firm 2 
0.188 
Firm 3 
0.462 
This portfolio is Portfolio A in Figure 6.1 but we can consider an alternative portfolio that also has a 13% target return. This portfolio is engineered by combining the riskfree security and the risky Portfolio I by the appropriate choice of a. The weights of 1/3 and 2/3 provide a target return of 13 1/3%. To reach our new target of 13% we need more weight on the riskfree component and less weight on the risky component. That is, we want to choose a so that a rf + (1 a)E(r) equals .13. Since E(r) equals .14 and rf equals .12, clearly a must equal .5. In Figure 6.1, this is denoted as Portfolio B.
We now have two portfolios that provide an expected return equal to 0.13. The question that remains is: Does one portfolio dominate the other?
Notice in Figure 6.1. the different slopes of the line drawn between the riskfree security point (zero standard deviation, and expected return = 0.12) and portfolios A and B, respectively. The line through B has a greater slope than the line through A. This is because the expected return from B is 0.14 and the expected return from A is 0.13. Every point on the line through B (apart from rf itself) lies above every point on the line through A. The portfolio variance for the portfolio of half riskfree asset and half Portfolio B must be smaller than the portfolio variance for the portfolio of zero riskfree asset and only Portfolio A. Therefore, Portfolio B dominates Portfolio A.
Numerically, this can be verified by computing the portfolio standard deviation for both portfolios. This is computed as:
The subscripts i,j are defined from 1 to 3 for the 3 securities.
For a review of this formula see Chapter 2, topic 2.9, Portfolio Statistics. Applied to the current example, the two portfolio standard deviations are:
Portfolio Standard Deviation 

Portfolio A 
0.1179 

Portfolio B(.5 rf .5I) 
0.0903 
As a result, the second portfolio that exploits the existence of the riskfree security dominates Portfolio A.
So what is the best that you can do? This question is explored in topic 6.6, Capital Market Line.
previous topic
next topic
(C) Copyright 1999, OS Financial Trading System